Rank and Nielsen equivalence in hyperbolic extensions

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of Out$(F_n)$.
Comment: v2: 10 pages. Minor updates to incorporate referee comments. Added counter-example demonstrating necessity of torsion free hypothesis. Final version; accepted for publication in the International Journal of Algebra and Computation (IJAC)